Using this, and the fact that both graphical and geometric small cancellations labellings exist (again, in the geometric case this is Gromov [1]}, or Arzhantseva–Delzant [2]} and in the graphical case this is Osajda [3]}) for certain families of finite graphs of large girth. A similar inductive setup can be used to produce injectivity with asymptotic controls - combining them at each stage would give both situations simultaneously.
